Trump said Kim has a "certain vision, it's not exactly our vision, but it's much closer than it was a year ago."
State Secretary Mike Pompeo told the press conference that he hopes an agreement will be reached "in the coming weeks."
He added, "We didn't get all the way. We asked him to do more, he was unprepared for it. I'm still optimistic."
The president also touched on Michael Cohen's Witness Congress Testimony Wednesday says it his former personal lawyer and fixer hadn't lied about everything.
Earlier, the White House press secretary, Sarah Sanders, published a statement that no consensus had been reached, but "the respective teams look forward to meeting in the future."
While Trump has said he was not busy entering into a comprehensive pact with Kim, the president told a "very strong partnership" with the North Korean leader before diverting Vietnam to Washington.
The president also said that Kim had promised "the test will not start" with rockets or missiles "or anything related to nuclear power."
The apparent collapse of the negotiations is sure to come as a relief to many North Korean experts – including some democratic and Republican legislators – who worried that Trump was ready to make concessions to Kim without securing a firm and verifiable disarmament commitment from the dictator.
Earlier Thursday, NBC News first reported that US officials entered the summit when they had lost their demand that Kim give full account of his nuclear weapons and ballistic missiles program.
In rare remarks to Western reporters before the summit was shortened, Kim, who spoke through an interpreter, said he was willing to consider "denuclearizing".
"If I'm not willing to do that, I wouldn't be here right now," Kim said as he sat over the Trump negotiating table.
But the word has been ambiguous in US negotiations with North Korea.
Trump himself repeatedly stressed that he was in the "no rush" to get a comprehensive deal with Kim and to shake off the expectations of a complete nuclear disarmament pact.
Kim said during his initial remarks in Hanoi that skeptics of the relationship would watch each other and see the two leaders "side by side as if they were watching a fancy movie".
Kim and Trump had a one-to-one session in the morning followed by a meeting with a larger group of officials from both sides. But everything fell apart around lunch time.
The two leaders had a number of other issues to discuss, including a possible statement on the end of the nearly 70-year-old Korean war, the possible removal of US forces from the Korean Peninsula and the lifting of the criminal and American economic sanctions designed for to force Pyongyang to abandon its weapons programs.
But it is complete verifiable disarmament of North Korea's nuclear program as the US side ultimately wants.
Before the summit ended unexpectedly, Kim was asked if he was sure he could find a deal with Trump.
"It's too early to say," Kim replied. "I didn't want to say I'm pessimistic."
Trump said he was willing to take his time – a necessity, it seems he will eventually persuade Kim to disarm.
"President Kim and myself, we will make the right deal," Trump said. "Speed is not important."
F. Brinley Bruton and Jason Cumming reported from London.